ephermal enforcement of import order to circumvent current problem in merging interpretation morphisms
1 (* Title: HOL/Relation.thy
2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
3 Copyright 1996 University of Cambridge
9 imports Finite_Set Datatype
10 (*FIXME order is important, otherwise merge problem for canonical interpretation of class monoid_mult wrt. power!*)
13 subsection {* Definitions *}
16 converse :: "('a * 'b) set => ('b * 'a) set"
17 ("(_^-1)" [1000] 999) where
18 "r^-1 == {(y, x). (x, y) : r}"
21 converse ("(_\<inverse>)" [1000] 999)
24 rel_comp :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set"
26 "r O s == {(x,z). EX y. (x, y) : s & (y, z) : r}"
29 Image :: "[('a * 'b) set, 'a set] => 'b set"
30 (infixl "``" 90) where
31 "r `` s == {y. EX x:s. (x,y):r}"
34 Id :: "('a * 'a) set" where -- {* the identity relation *}
35 "Id == {p. EX x. p = (x,x)}"
38 Id_on :: "'a set => ('a * 'a) set" where -- {* diagonal: identity over a set *}
39 "Id_on A == \<Union>x\<in>A. {(x,x)}"
42 Domain :: "('a * 'b) set => 'a set" where
43 "Domain r == {x. EX y. (x,y):r}"
46 Range :: "('a * 'b) set => 'b set" where
47 "Range r == Domain(r^-1)"
50 Field :: "('a * 'a) set => 'a set" where
51 "Field r == Domain r \<union> Range r"
54 refl_on :: "['a set, ('a * 'a) set] => bool" where -- {* reflexivity over a set *}
55 "refl_on A r == r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)"
58 refl :: "('a * 'a) set => bool" where -- {* reflexivity over a type *}
59 "refl == refl_on UNIV"
62 sym :: "('a * 'a) set => bool" where -- {* symmetry predicate *}
63 "sym r == ALL x y. (x,y): r --> (y,x): r"
66 antisym :: "('a * 'a) set => bool" where -- {* antisymmetry predicate *}
67 "antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y"
70 trans :: "('a * 'a) set => bool" where -- {* transitivity predicate *}
71 "trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
74 irrefl :: "('a * 'a) set => bool" where
75 "irrefl r \<equiv> \<forall>x. (x,x) \<notin> r"
78 total_on :: "'a set => ('a * 'a) set => bool" where
79 "total_on A r \<equiv> \<forall>x\<in>A.\<forall>y\<in>A. x\<noteq>y \<longrightarrow> (x,y)\<in>r \<or> (y,x)\<in>r"
81 abbreviation "total \<equiv> total_on UNIV"
84 single_valued :: "('a * 'b) set => bool" where
85 "single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)"
88 inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set" where
89 "inv_image r f == {(x, y). (f x, f y) : r}"
92 subsection {* The identity relation *}
94 lemma IdI [intro]: "(a, a) : Id"
97 lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
98 by (unfold Id_def) (iprover elim: CollectE)
100 lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
101 by (unfold Id_def) blast
103 lemma refl_Id: "refl Id"
104 by (simp add: refl_on_def)
106 lemma antisym_Id: "antisym Id"
107 -- {* A strange result, since @{text Id} is also symmetric. *}
108 by (simp add: antisym_def)
110 lemma sym_Id: "sym Id"
111 by (simp add: sym_def)
113 lemma trans_Id: "trans Id"
114 by (simp add: trans_def)
117 subsection {* Diagonal: identity over a set *}
119 lemma Id_on_empty [simp]: "Id_on {} = {}"
120 by (simp add: Id_on_def)
122 lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A"
123 by (simp add: Id_on_def)
125 lemma Id_onI [intro!,noatp]: "a : A ==> (a, a) : Id_on A"
126 by (rule Id_on_eqI) (rule refl)
128 lemma Id_onE [elim!]:
129 "c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
130 -- {* The general elimination rule. *}
131 by (unfold Id_on_def) (iprover elim!: UN_E singletonE)
133 lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)"
136 lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
140 subsection {* Composition of two relations *}
142 lemma rel_compI [intro]:
143 "(a, b) : s ==> (b, c) : r ==> (a, c) : r O s"
144 by (unfold rel_comp_def) blast
146 lemma rel_compE [elim!]: "xz : r O s ==>
147 (!!x y z. xz = (x, z) ==> (x, y) : s ==> (y, z) : r ==> P) ==> P"
148 by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE)
151 "(a, c) : r O s ==> (!!y. (a, y) : s ==> (y, c) : r ==> P) ==> P"
152 by (iprover elim: rel_compE Pair_inject ssubst)
154 lemma R_O_Id [simp]: "R O Id = R"
157 lemma Id_O_R [simp]: "Id O R = R"
160 lemma rel_comp_empty1[simp]: "{} O R = {}"
163 lemma rel_comp_empty2[simp]: "R O {} = {}"
166 lemma O_assoc: "(R O S) O T = R O (S O T)"
169 lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
170 by (unfold trans_def) blast
172 lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
175 lemma rel_comp_subset_Sigma:
176 "s \<subseteq> A \<times> B ==> r \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
179 lemma rel_comp_distrib[simp]: "R O (S \<union> T) = (R O S) \<union> (R O T)"
182 lemma rel_comp_distrib2[simp]: "(S \<union> T) O R = (S O R) \<union> (T O R)"
186 subsection {* Reflexivity *}
188 lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"
189 by (unfold refl_on_def) (iprover intro!: ballI)
191 lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r"
192 by (unfold refl_on_def) blast
194 lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A"
195 by (unfold refl_on_def) blast
197 lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"
198 by (unfold refl_on_def) blast
200 lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"
201 by (unfold refl_on_def) blast
203 lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)"
204 by (unfold refl_on_def) blast
207 "ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)"
208 by (unfold refl_on_def) fast
211 "ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"
212 by (unfold refl_on_def) blast
214 lemma refl_on_empty[simp]: "refl_on {} {}"
215 by(simp add:refl_on_def)
217 lemma refl_on_Id_on: "refl_on A (Id_on A)"
218 by (rule refl_onI [OF Id_on_subset_Times Id_onI])
221 subsection {* Antisymmetry *}
224 "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
225 by (unfold antisym_def) iprover
227 lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
228 by (unfold antisym_def) iprover
230 lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
231 by (unfold antisym_def) blast
233 lemma antisym_empty [simp]: "antisym {}"
234 by (unfold antisym_def) blast
236 lemma antisym_Id_on [simp]: "antisym (Id_on A)"
237 by (unfold antisym_def) blast
240 subsection {* Symmetry *}
242 lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
243 by (unfold sym_def) iprover
245 lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
246 by (unfold sym_def, blast)
248 lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
249 by (fast intro: symI dest: symD)
251 lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)"
252 by (fast intro: symI dest: symD)
254 lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)"
255 by (fast intro: symI dest: symD)
257 lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)"
258 by (fast intro: symI dest: symD)
260 lemma sym_Id_on [simp]: "sym (Id_on A)"
261 by (rule symI) clarify
264 subsection {* Transitivity *}
267 "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
268 by (unfold trans_def) iprover
270 lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
271 by (unfold trans_def) iprover
273 lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
274 by (fast intro: transI elim: transD)
276 lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)"
277 by (fast intro: transI elim: transD)
279 lemma trans_Id_on [simp]: "trans (Id_on A)"
280 by (fast intro: transI elim: transD)
282 lemma trans_diff_Id: " trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r-Id)"
283 unfolding antisym_def trans_def by blast
285 subsection {* Irreflexivity *}
287 lemma irrefl_diff_Id[simp]: "irrefl(r-Id)"
288 by(simp add:irrefl_def)
290 subsection {* Totality *}
292 lemma total_on_empty[simp]: "total_on {} r"
293 by(simp add:total_on_def)
295 lemma total_on_diff_Id[simp]: "total_on A (r-Id) = total_on A r"
296 by(simp add: total_on_def)
298 subsection {* Converse *}
300 lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
301 by (simp add: converse_def)
303 lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
304 by (simp add: converse_def)
306 lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
307 by (simp add: converse_def)
309 lemma converseE [elim!]:
310 "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
311 -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
312 by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
314 lemma converse_converse [simp]: "(r^-1)^-1 = r"
315 by (unfold converse_def) blast
317 lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
320 lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
323 lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
326 lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
329 lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
332 lemma converse_Id [simp]: "Id^-1 = Id"
335 lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A"
338 lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
339 by (unfold refl_on_def) auto
341 lemma sym_converse [simp]: "sym (converse r) = sym r"
342 by (unfold sym_def) blast
344 lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
345 by (unfold antisym_def) blast
347 lemma trans_converse [simp]: "trans (converse r) = trans r"
348 by (unfold trans_def) blast
350 lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
351 by (unfold sym_def) fast
353 lemma sym_Un_converse: "sym (r \<union> r^-1)"
354 by (unfold sym_def) blast
356 lemma sym_Int_converse: "sym (r \<inter> r^-1)"
357 by (unfold sym_def) blast
359 lemma total_on_converse[simp]: "total_on A (r^-1) = total_on A r"
360 by (auto simp: total_on_def)
363 subsection {* Domain *}
365 declare Domain_def [noatp]
367 lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
368 by (unfold Domain_def) blast
370 lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
371 by (iprover intro!: iffD2 [OF Domain_iff])
373 lemma DomainE [elim!]:
374 "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
375 by (iprover dest!: iffD1 [OF Domain_iff])
377 lemma Domain_empty [simp]: "Domain {} = {}"
380 lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
383 lemma Domain_Id [simp]: "Domain Id = UNIV"
386 lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
389 lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
392 lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
395 lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
398 lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
401 lemma Domain_converse[simp]: "Domain(r^-1) = Range r"
402 by(auto simp:Range_def)
404 lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
407 lemma fst_eq_Domain: "fst ` R = Domain R";
408 by (auto intro!:image_eqI)
410 lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
413 lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
417 subsection {* Range *}
419 lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
420 by (simp add: Domain_def Range_def)
422 lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
423 by (unfold Range_def) (iprover intro!: converseI DomainI)
425 lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
426 by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
428 lemma Range_empty [simp]: "Range {} = {}"
431 lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
434 lemma Range_Id [simp]: "Range Id = UNIV"
437 lemma Range_Id_on [simp]: "Range (Id_on A) = A"
440 lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
443 lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
446 lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
449 lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
452 lemma Range_converse[simp]: "Range(r^-1) = Domain r"
455 lemma snd_eq_Range: "snd ` R = Range R";
456 by (auto intro!:image_eqI)
459 subsection {* Field *}
461 lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
462 by(auto simp:Field_def Domain_def Range_def)
464 lemma Field_empty[simp]: "Field {} = {}"
465 by(auto simp:Field_def)
467 lemma Field_insert[simp]: "Field (insert (a,b) r) = {a,b} \<union> Field r"
468 by(auto simp:Field_def)
470 lemma Field_Un[simp]: "Field (r \<union> s) = Field r \<union> Field s"
471 by(auto simp:Field_def)
473 lemma Field_Union[simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
474 by(auto simp:Field_def)
476 lemma Field_converse[simp]: "Field(r^-1) = Field r"
477 by(auto simp:Field_def)
480 subsection {* Image of a set under a relation *}
482 declare Image_def [noatp]
484 lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
485 by (simp add: Image_def)
487 lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
488 by (simp add: Image_def)
490 lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
491 by (rule Image_iff [THEN trans]) simp
493 lemma ImageI [intro,noatp]: "(a, b) : r ==> a : A ==> b : r``A"
494 by (unfold Image_def) blast
496 lemma ImageE [elim!]:
497 "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
498 by (unfold Image_def) (iprover elim!: CollectE bexE)
500 lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
501 -- {* This version's more effective when we already have the required @{text a} *}
504 lemma Image_empty [simp]: "R``{} = {}"
507 lemma Image_Id [simp]: "Id `` A = A"
510 lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
513 lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
517 "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
518 by (simp add: single_valued_def, blast)
520 lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
523 lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
526 lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
527 by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
529 lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
530 -- {* NOT suitable for rewriting *}
533 lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
536 lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
539 lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
542 text{*Converse inclusion requires some assumptions*}
544 "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
545 apply (rule equalityI)
546 apply (rule Image_INT_subset)
547 apply (simp add: single_valued_def, blast)
550 lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
554 subsection {* Single valued relations *}
556 lemma single_valuedI:
557 "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
558 by (unfold single_valued_def)
560 lemma single_valuedD:
561 "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
562 by (simp add: single_valued_def)
564 lemma single_valued_rel_comp:
565 "single_valued r ==> single_valued s ==> single_valued (r O s)"
566 by (unfold single_valued_def) blast
568 lemma single_valued_subset:
569 "r \<subseteq> s ==> single_valued s ==> single_valued r"
570 by (unfold single_valued_def) blast
572 lemma single_valued_Id [simp]: "single_valued Id"
573 by (unfold single_valued_def) blast
575 lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
576 by (unfold single_valued_def) blast
579 subsection {* Graphs given by @{text Collect} *}
581 lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
584 lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
587 lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
591 subsection {* Inverse image *}
593 lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
594 by (unfold sym_def inv_image_def) blast
596 lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
597 apply (unfold trans_def inv_image_def)
598 apply (simp (no_asm))
603 subsection {* Finiteness *}
605 lemma finite_converse [iff]: "finite (r^-1) = finite r"
606 apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
609 apply (erule finite_imageD [unfolded inj_on_def])
610 apply (simp split add: split_split)
611 apply (erule finite_imageI)
612 apply (simp add: converse_def image_def, auto)
614 prefer 2 apply assumption
618 text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
621 lemma finite_Field: "finite r ==> finite (Field r)"
622 -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
623 apply (induct set: finite)
624 apply (auto simp add: Field_def Domain_insert Range_insert)
628 subsection {* Version of @{text lfp_induct} for binary relations *}
631 lfp_induct_set [of "(a, b)", split_format (complete)]